有界d维域上分数阶泊松方程的球上行走驱动有限差分方法

IF 2.3 2区 数学 Q1 MATHEMATICS, APPLIED
Daxin Nie, Jing Sun, Weihua Deng
{"title":"有界d维域上分数阶泊松方程的球上行走驱动有限差分方法","authors":"Daxin Nie, Jing Sun, Weihua Deng","doi":"10.1093/imanum/draf031","DOIUrl":null,"url":null,"abstract":"Inspired by the idea of ‘walk-on-sphere’ algorithm, we propose a novel finite-difference framework for solving the fractional Poisson equation under the help of the Feynman-Kac representation of its solution, i.e., walk-on-sphere-motivated finite-difference scheme. By choosing suitable basis functions in interpolatory quadrature and using graded meshes, the convergence rates can achieve up to $O(h^{2})$ in arbitrary $d$-dimensional bounded Lipschitz domain satisfying the exterior ball condition, where $d>1$; while the convergence rate can reach $O(h^{10})$ in 1-dimensional bounded domain under some regularity assumptions on the source term $f$. Furthermore, we propose a strict convergence analysis and several numerical examples in different domains, including circle, L-shape, pentagram and ball, are provided to illustrate the effectiveness of the above built scheme.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"8 1","pages":""},"PeriodicalIF":2.3000,"publicationDate":"2025-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A walk-on-sphere-motivated finite-difference method for the fractional Poisson equation on a bounded d-dimensional domain\",\"authors\":\"Daxin Nie, Jing Sun, Weihua Deng\",\"doi\":\"10.1093/imanum/draf031\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Inspired by the idea of ‘walk-on-sphere’ algorithm, we propose a novel finite-difference framework for solving the fractional Poisson equation under the help of the Feynman-Kac representation of its solution, i.e., walk-on-sphere-motivated finite-difference scheme. By choosing suitable basis functions in interpolatory quadrature and using graded meshes, the convergence rates can achieve up to $O(h^{2})$ in arbitrary $d$-dimensional bounded Lipschitz domain satisfying the exterior ball condition, where $d>1$; while the convergence rate can reach $O(h^{10})$ in 1-dimensional bounded domain under some regularity assumptions on the source term $f$. Furthermore, we propose a strict convergence analysis and several numerical examples in different domains, including circle, L-shape, pentagram and ball, are provided to illustrate the effectiveness of the above built scheme.\",\"PeriodicalId\":56295,\"journal\":{\"name\":\"IMA Journal of Numerical Analysis\",\"volume\":\"8 1\",\"pages\":\"\"},\"PeriodicalIF\":2.3000,\"publicationDate\":\"2025-05-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"IMA Journal of Numerical Analysis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1093/imanum/draf031\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"IMA Journal of Numerical Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1093/imanum/draf031","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0

摘要

受“walk-on-sphere”算法思想的启发,我们提出了一种新的有限差分框架,在其解的费曼-卡茨表示的帮助下求解分数阶泊松方程,即walk-on-sphere-motivated有限差分格式。通过在插值正交中选择合适的基函数并使用梯度网格,在满足外球条件的任意d维有界Lipschitz域中,收敛速率可达到$O(h^{2})$,其中$d>1$;在源项$f$的一些正则性假设下,在一维有界域中收敛速度可达$O(h^{10})$。此外,我们给出了严格的收敛性分析,并给出了在圆、l形、五角星和球等不同区域的数值算例,以说明所建方案的有效性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A walk-on-sphere-motivated finite-difference method for the fractional Poisson equation on a bounded d-dimensional domain
Inspired by the idea of ‘walk-on-sphere’ algorithm, we propose a novel finite-difference framework for solving the fractional Poisson equation under the help of the Feynman-Kac representation of its solution, i.e., walk-on-sphere-motivated finite-difference scheme. By choosing suitable basis functions in interpolatory quadrature and using graded meshes, the convergence rates can achieve up to $O(h^{2})$ in arbitrary $d$-dimensional bounded Lipschitz domain satisfying the exterior ball condition, where $d>1$; while the convergence rate can reach $O(h^{10})$ in 1-dimensional bounded domain under some regularity assumptions on the source term $f$. Furthermore, we propose a strict convergence analysis and several numerical examples in different domains, including circle, L-shape, pentagram and ball, are provided to illustrate the effectiveness of the above built scheme.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
IMA Journal of Numerical Analysis
IMA Journal of Numerical Analysis 数学-应用数学
CiteScore
5.30
自引率
4.80%
发文量
79
审稿时长
6-12 weeks
期刊介绍: The IMA Journal of Numerical Analysis (IMAJNA) publishes original contributions to all fields of numerical analysis; articles will be accepted which treat the theory, development or use of practical algorithms and interactions between these aspects. Occasional survey articles are also published.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信